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Connecting with Computer Science, 2e

Connecting with Computer Science, 2e. Chapter 7 Numbering Systems and Data Representations. Objectives. In this chapter you will: Learn why numbering systems are important to understand Refresh your knowledge of powers of numbers Learn how numbering systems are used to count

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Connecting with Computer Science, 2e

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  1. Connecting with Computer Science, 2e Chapter 7 Numbering Systems and Data Representations

  2. Objectives • In this chapter you will: • Learn why numbering systems are important to understand • Refresh your knowledge of powers of numbers • Learn how numbering systems are used to count • Understand the significance of positional value in a numbering system • Learn the differences and similarities between numbering system bases Connecting with Computer Science, 2e

  3. Objectives (cont’d.) • In this chapter you will (cont’d.): • Learn how to convert numbers between bases • Learn how to do binary and hexadecimal math • Learn how data is represented as binary in a computer • Learn how images and sounds are stored in a computer Connecting with Computer Science, 2e

  4. Why You Need to Know About...Numbering Systems • Computers store programs and data as binary digits • Hexadecimal number system • Provides convenient representation • Written into error messages • Helpful as students interact with computers in later computing courses and throughout their careers Connecting with Computer Science, 2e

  5. Powers of Numbers: A Refresher • Raising a number to a positive power (exponent) • Self-multiply number by the specified power • Example: 23 = 2 * 2 * 2 = 8 (asterisk = multiplication) • Special cases: 0 and 1 as powers • Any number raised to 0 = 1: 10,5550 = 1 • Any number raised to 1 = itself: 10,5551 = 10,555 • Raising a number to a negative power • Follow same steps for positive power • Divide result into one • Example: 2-3 = 1 / (23) = .125 Connecting with Computer Science, 2e

  6. Counting Things • Numbers are used to count things • Base 10 (decimal): most familiar • Computer uses base 2 (binary) • Two unique digits: 0 and 1 • Base 16 (hexadecimal) represent binary digits • Sixteen unique digits: 0–9, A–F • Counting in all number systems • Count digits defined in number system until exhausted • Place 0 in ones column and carry 1 to the left Connecting with Computer Science, 2e

  7. Positional Value • Key principle of numbering systems • Weight assigned to digit • Based on position in number • Steps for base 10 • Determine positional value of each digit by raising 10 to position within number • Radix point • Divides fractional portion from the whole portion of a number Connecting with Computer Science, 2e

  8. Positional Value (cont’d.) • Positional value example: 436.95 • 4 in position two • 3 in position one • 6 in position zero • To the right of the radix point: • 9 in position negative one • 5 in position negative two Connecting with Computer Science, 2e

  9. Positional Value (cont’d.) • Positional value specifies what multiplier the position gives to the overall number • Example: 4321 • Digit 3 multiplied by positional value of its position: 100 (102) • Sum of all digits multiplied by positional value determines total number of things being counted Connecting with Computer Science, 2e

  10. Positional Value (cont’d.) Figure 7-1, Positional values for a base 10 number Connecting with Computer Science, 2e

  11. Positional Value (cont’d.) • Positional value of the binary number 10112 • Rightmost position: positional value of the base (2) raised to the 0 power • Positional value: 1 • Next position: value of 2 raised to the power of 1 • Next position: 2 squared • Next position: 2 to the third • Positional value significance • Gives weight each digit contributes to the number’s overall value Connecting with Computer Science, 2e

  12. Positional Value (cont’d.) Figure 7-2, Positional values for a base 2 number Connecting with Computer Science, 2e

  13. How Many Things Does a Number Represent? • Number equals sum of each digit x positional value • Translate with base 10 • Example: 10012 • Equivalent to nine things • (1 * 20) + (0 * 21) + (0 * 22) + (1 * 23) • 1 + 0 + 0 + 8 = 9 • General procedure (any base) • Calculate position value of the number by raising the base value to the power of the position • Multiply positional value by digit in that position • Add each calculated value together Connecting with Computer Science, 2e

  14. Converting Numbers Between Bases • Can represent any quantity in any base • Counting process similar for all bases • Count until highest digit for base reached • Add 1 to next higher position to left • Return to 0 in current position • Conversion: map from one base to another • Identities easily calculated • Identities obtained by table lookup Connecting with Computer Science, 2e

  15. Converting Numbers Between Bases (cont’d.) Table 7-1, Counting in different bases Connecting with Computer Science, 2e

  16. Converting to Base 10 • Methods: • Table lookup (more extensive than Table 7-1) • Algorithm for evaluating number in any base • Multiply the digit in each position by its positional value • Add results • Example: 1316 • Identify base: 16 • Map positions to digits • Raise, multiply, and add: • 1316 = (3 x 160) + (1 x 161) = 19 Connecting with Computer Science, 2e

  17. Converting from Base 10 • Method: • Reverse of converting to base 10 • Follow an algorithm for converting from base 10 • Algorithm • Determine target base positional value nearest to or equal in value to decimal number • Determine how many times that positional value can be divided into the decimal number • Multiply number from Step 2 by the associated positional value, and subtract product from number chosen Connecting with Computer Science, 2e

  18. Converting from Base 10 (cont’d.) • Algorithm (cont’d.) • Use remainder from Step 3 as a new starting value, and repeat Steps 1-3 until Step 1 value is less than or equal to the maximum value in the target base • Converted number is the digits written down, in order from left to right Connecting with Computer Science, 2e

  19. Binary and Hexadecimal Math • Procedure for adding numbers similar in all bases • Difference lies in carry process • Value of carry equals value of base • Procedure for subtraction • Similar Connecting with Computer Science, 2e

  20. Binary and Hexadecimal Math (cont’d.) Figure 7-3, Adding numbers in binary Connecting with Computer Science, 2e

  21. Binary and Hexadecimal Math (cont’d.) Figure 7-4, Subtraction with base 2 numbers Connecting with Computer Science, 2e

  22. Data Representation in Binary • Binary values map to two states • On or off • Bit • Each 1 and 0 (on and off ) in a computer • Byte • Group of 8 bits • Word • Collection of bytes (typically 4 bytes) • Nibble • Half a byte or 4 bits Connecting with Computer Science, 2e

  23. Data Representation in Binary (cont’d.) • Hexadecimal used as binary shorthand • Relate each hexadecimal digit to 4-bit binary pattern • Example: 1111 1010 1100 1110 = F A C E • See Table 7-1 for verification • Example: C2D4 = 1100001011010100 • Information helps debug error messages Connecting with Computer Science, 2e

  24. Representing Whole Numbers • Whole numbers (integer numbers) • Stored in fixed number of bits • 2010 stored as 16-bit integer 0000011111011010 • Equivalent hex value: 07DA • Signed numbers stored with twos complement • Leftmost bit reserved for sign • 1 = negative and 0 = positive • If positive: leave as is • If negative: perform twos complement • Reverse bit pattern and add 1 to number using binary addition Connecting with Computer Science, 2e

  25. Representing Whole Numbers (cont’d.) Figure 7-5, Storing numbers in a twos complement 8-bit field Connecting with Computer Science, 2e

  26. Representing Fractional Numbers • Computers store fractional numbers • Negative and positive • Storage technique based on floating-point notation • Example: 1.345E+5 • 1.345 = mantissa, E = exponent, + 5 moves decimal • IEEE-754 specification • Uses binary mantissas and exponents • Implementation details are part of advanced study Connecting with Computer Science, 2e

  27. Representing Characters • Computers store characters according to standards • ASCII • Represents characters with 7-bit pattern • Provides for upper- and lowercase English letters, numeric characters, punctuation, special characters • Accommodates 128 (27) different characters • Globalization places upward pressure • Extended ASCII: allows 8-bit patterns (256 total) • Unicode: defined for 16-bit patterns (34,168 total) Connecting with Computer Science, 2e

  28. Representing Characters (cont’d.) Table 7-2, Sample standard ASCII characters Connecting with Computer Science, 2e

  29. Representing Images • Screen image made up of small dots of colored light • Dot called “pixel” (picture element): smallest unit • Resolution: # pixels in each row and column • Each pixel stored in the computer as a binary pattern • RGB encoding • Red, blue, and green assigned to eight of 24 bits • White represented with 1s, black with 0s • Color: amount of red, green, and blue specified in each of the 8-bit sections Connecting with Computer Science, 2e

  30. Representing Images (cont’d.) • Images are stored with pixel-based technologies • Compress large image files • JPG • GIF • Compress moving images • WAV • MP3 Connecting with Computer Science, 2e

  31. Representing Sounds • Sound represented as waveform • Amplitude (volume) • Frequency (pitch) • Computer samples sounds at fixed intervals • Samples given a binary value according to amplitude • Bits in each sample determine amplitude range • CD-quality audio • Sound must be sampled over 44,000 times a second • Samples must allow > 65,000 different amplitudes Connecting with Computer Science, 2e

  32. Representing Sounds (cont’d.) Figure 7-6, Digital sampling of a sound wave Connecting with Computer Science, 2e

  33. One Last Thought • To excel in the computer field: • Student must understand number conversions and data representations • Everything stored in or takes place on a computer • Ultimately done in binary • Computer professionals with strong understanding of chapter concepts will: • Perform better at whatever they specialize in • Become more essential in their organizations Connecting with Computer Science, 2e

  34. Summary • Knowledge of alternative number systems essential • Machine language based on binary system • Hexadecimal used to represent binary numbers • Any number can be represented in any base • Positional value: weight based on digit position • Counting processes similar for all bases • Conversion between bases: one-to-one mapping • Arithmetic is defined for all bases • Data representation: bits, nibbles, bytes, words Connecting with Computer Science, 2e

  35. Summary (cont’d.) • Twos complement is a technique for storing signed numbers • Floating-point notation is a system used to represent fractions • ASCII and Unicode: character set standards • Image representation: based on binary pixels • Sound representation: based on amplitude samples Connecting with Computer Science, 2e

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